We now construct an example which interpolates Example 5.5 and 5.7. Consider the smooth function:

$$H_{0}=\frac{\pi}{4}\im(u_{1}\overline{u}_{2})$$

and let $\eta_{H_{0}}$ be the Hamiltonian vector field associated to $H_{0}$. If $\Phi_{s}$ is the flow generated by $\eta_{H_{0}}$, then the Hamiltonian symplectomorphism associated to $H_{0}$ is defined to be $\Phi_{H_{0}}=\Phi_{1}$. One computes that in our case

$$\Phi_{H_{0}}:(u_{1},u_{2})\mapsto\frac{1}{\sqrt{2}}(u_{1}-u_{2},u_{1}+u_{2}).$$

It maps $\{u_{1}=0\}$ to $\{v_{1}+v_{2}=0\}$. We now want a symplectomorphism which acts like $\Phi_{H_{0}}$ in a small ball centered at the origin and like the identity outside a slightly bigger ball. So choose a cut-off function $k:\mathbb{R}_{\geq 0}\rightarrow[0,1]$ such that, for some $\epsilon>0$,

$$k(t)=\left\{\begin{array}[]{l}1\ \quad\text{when}\ 0<t\leq\epsilon;\\ 0\ \quad\text{when}\ t\geq 2\epsilon\end{array}\right.$$

(45)

and define the Hamiltonian

$$H=k(|u_{1}|^{2}+|u_{2}|^{2})H_{0}.$$

The Hamiltonian symplectomorphism $\Phi_{H}$ associated to $H$ satisfies

$$\Phi_{H}(u_{1},u_{2})=\begin{cases}\id_{\mathbb{C}^{2}},&\quad\text{when}\ |u_{1}|^{2}+|u_{2}|^{2}\geq 2\epsilon;\\ \\ \frac{1}{\sqrt{2}}(u_{1}-u_{2},u_{1}+u_{2}),&\quad\text{when}\ |u_{1}|^{2}+|u_{2}|^{2}\leq\epsilon.\end{cases}$$

Now let $\Psi$ be the affine symplectomorphism

$$\Psi:(v_{1},v_{2})\mapsto\frac{1}{\sqrt{2}}(v_{1}-v_{2},v_{1}+v_{2}-\sqrt{2}).$$

and finally, define $\Phi=\Psi\circ\Phi_{H}$. It is clear that

$$\Phi(u_{1},u_{2})=\begin{cases}\Psi,&\quad\text{when}\ |u_{1}|^{2}+|u_{2}|^{2}\geq 2\epsilon;\\ \\ (-u_{2},u_{1}-1),&\quad\text{when}\ |u_{1}|^{2}+|u_{2}|^{2}\leq\epsilon.\end{cases}$$

Notice that $\Phi$ acts like in (37) on the ball of radius $\sqrt{\epsilon}$ around the origin, i.e. in a neighborhood of the surface $\{u_{1}=0\}$, and like in (34) outside a larger ball. We use this $\Phi$ to construct a fibration $f$ using Proposition 5.4. One can then see that $\Phi\circ\Gamma_{0}$ sends $\Sigma$ to a surface $\Sigma^{\prime}$ such that $\Log(\Sigma^{\prime})\subset\mathbb{R}^{2}$ is a 3-legged amoeba with the end of the horizontal leg pinched down to a straight line. The discriminant locus of $f$ is then $\Delta=\{0\}\times\Log(\Sigma^{\prime})\subset\mathbb{R}^{3}$. Of course, $f$ fails to be smooth on the slice $\mu^{-1}(0)$. Using the same method we can twist $\Sigma$ suitably and obtain a fibrations having discriminant locus an amoeba with three thin legs (cf. Figure 5). For example, to pinch the diagonal leg to a thin line, choose a smooth function $H_{0}$ generating the Hamiltonian symplectomorphism

$$(u_{1},u_{2})\rightarrow\left(u_{1}+\frac{1}{\sqrt{2}},u_{2}+\frac{1}{\sqrt{2}}\right).$$

Cut $H_{0}$ off with a function $\rho$ which vanishes when $|u_{2}|^{2}\leq M/2$, for some big $M$, and is equal to $1$ when $|u_{2}|^{2}\geq M$. This produces a Hamiltonian $H$. Now one proceeds as before. With an almost identical procedure one pinches down the vertical leg. The final choice of symplectomorphism $\Phi$ pinching down all three legs simultaneously may look like:

$$\Phi(u_{1},u_{2})=\begin{cases}(-u_{2},u_{1}-1),&\quad\text{when}\ |u_{1}|^{2}+|u_{2}|^{2}\leq\epsilon;\\ \\ (u_{1}-1,u_{2}-\sqrt{2}),&\quad\text{when}\ |u_{1}|^{2}+|u_{2}-\sqrt{2}|^{2}\leq\epsilon;\\ \\ \frac{1}{\sqrt{2}}(u_{1}-u_{2},\,u_{1}+u_{2}),&\quad\text{when}\ |u_{2}|^{2}\geq M;\\ \\ \Psi,&\quad\text{everywhere else}.\end{cases}$$

(46)

It is clear that this piecewise smooth example is topologically conjugate to the one in Example 2.9. Here we have made explicit the twistings described there. In §7 we will show that this fibration can be modified so that it is actually smooth towards the ends of the three legs. For this we will develop further the smoothing method sketched at the end of Example 5.7. Also in §7, we will show that this fibration can be modified so that it is smooth away from a neighborhood homeomorphic to a 2-disk containing the codimension 1 part of its discriminant.